I started recently learning the newsvendor model, where V is the sell price, c is the purchase cost, h is the overage cost, π is the underage cost, Q is the quantity stocked, D is the demand for the product,
$ P(Q,D) = \begin{cases} (V-c)*Q-π(d-Q), & \text{d>Q} \\ V*d-c*Q+h(Q-d), & \text{d$\leq$Q} \end{cases}$
is the profit function,
$E(P(Q,D))=\int_0^\infty P(Q,D)*f_D(d) \,dd=\int_0^Q ((V-c)*Q-π(d-Q))*f_D(d) \,dd+\int_Q^\infty (V*d-c*Q+h(Q-d))*f_D(d) \,dd$
is the profit expectancy
Now,If D is distributed normally ($D \sim \mathcal{N}(μ,σ^2)$) and the optimal quantity of the product-$Q^*$ is known, how do I find the profit expectancy-E(P(Q,D)), solving the both integrals ( $\int_0^Q ((V-c)*Q-π(d-Q))*f_D(d) \,dd$ and $\int_Q^\infty (V*d-c*Q+h(Q-d))*f_D(d) \,dd)$ )?
I've looked it up in a lot of sources, and couldn't find a proper answer. Would appreciate any help.